where \(\hat{\mathcal {P}}_{\!\scriptscriptstyle AB}\)  is a parity-swap that sends
\( \left|x\right\rangle _{\!\scriptscriptstyle B} \otimes \left|y\right\rangle _{\!\scriptscriptstyle C}\)  to
\(\left|-x\right\rangle _{\!\scriptscriptstyle A} \otimes \left|y\right\rangle _{\!\scriptscriptstyle C}\) ,
mapping from
\({\mathcal {H}}_{\!\scriptscriptstyle B}^{\!\scriptscriptstyle (A)}\otimes {\mathcal {H}}_{\!\scriptscriptstyle C}^{\!\scriptscriptstyle (A)}\) ,
the Hilbert space for states of the composite system \(BC\) 
as described from \(A\) , to
\({\mathcal {H}}_{\!\scriptscriptstyle A}^{\!\scriptscriptstyle (B)}\otimes {\mathcal {H}}_{\!\scriptscriptstyle C}^{\!\scriptscriptstyle (B)}\) ,
the Hilbert space for states of the composite system \(AC\) 
as described from \(B\) , is given to achieve the above
operator transformations as
\({\mathcal {O}} \rightarrow \hat{S}_x {\mathcal {O}} \hat{S}_x^\dag \) 
[1]}. Given the fact that
\([\hat{x}_{\!\scriptscriptstyle I}^{\scriptscriptstyle (A)},\hat{p}_{\!\scriptscriptstyle J}^{\scriptscriptstyle (A)}] = \delta _{\!\scriptscriptstyle I\!J}i\) ,
(\(\hbar =1\)  units used throughout the paper), we have
\(e^{i \hat{x}_{\!\scriptscriptstyle B}^{\scriptscriptstyle (A)} \hat{p}_{\!\scriptscriptstyle C}^{\scriptscriptstyle (A)}}\) 
naively behaves as a translation in \(\hat{x}_{\!\scriptscriptstyle C}^{\scriptscriptstyle (A)}\) 
by the `parameter' \(\hat{x}_{\!\scriptscriptstyle B}^{\scriptscriptstyle (A)}\)  and as
a translation in \(\hat{p}_{\!\scriptscriptstyle B}^{\scriptscriptstyle (A)}\)  by the `parameter'
\(-\hat{p}_{\!\scriptscriptstyle C}^{\!\scriptscriptstyle (A)}\)  and the subsequent action of
\(\hat{\mathcal {P}}_{\!\scriptscriptstyle AB}\)  finishes the job.
